关于Fourier部分和算子范数上下界的新估值

A new evaluation for the norm of the Fourier operator S-n

研究Fourier算子Sn的范数‖Sn‖=1π∫π-πsin2n+12t2sint2dt.已知‖Sn‖具有表达式‖Sn‖=4π2logn+An,其中An表示与n相关且对n一致有界的数列.至今最好的估计是Rivlin给出的:‖Sn‖≤4π2logn+3,通过进一步精细的估计证明了4π2logn+1<‖Sn‖<4π2logn+2,从而给出了关于一致有界量An的上下界的一个新估计.

The norm of the Fourier operator S-n,‖S-n‖=1π∫+π-{-π}sin 2n+12t2sint2dt was studied.It is well known that ‖S-n‖=4π+2log n+A-n,where A-n is a uniformly bounded sequence.The best known evaluation was obtained by Rivlin:‖S-n‖≤4π+2log n+3.Through a precise evaluation,it was shown that {4π+2log n+1<‖S-n‖<4π+2log n+2.}